Ambitoric geometry I: Einstein metrics and extremal ambikähler structures

B.1 Conformal Killing objects

Let (M,c) be a conformal manifold. Among the conformally invariant linear differential operators on M, there is a family which are overdetermined of finite type, sometimes known as twistor or Penrose operators; their kernels are variously called twistors, tractors, or other names in special cases. Among the examples where the operator is first order are the equations for twistor forms (also known as conformal Killing forms) and conformal Killing tensors, both of which include conformal vector fields as a special case. There is also a second order equation for Einstein metrics in the conformal class. Apart from the obvious presence of (conformal) Killing vector fields and Einstein metrics, conformal Killing 2-tensors and twistor 2-forms are very relevant to the present work.

Let S0k⁢T⁢M denote the bundle of symmetric (0,k)-tensors 𝒮0 which are tracefree with respect to c in the sense that

for any conformal coframe εi. In particular, for k=2, 𝒮0∈S02⁢T⁢M may be identified with σ0∈L2⊗Sym0⁢(T⁢M) via

for any 1-form α and vector field X. Here Sym0⁢(T⁢M) is the bundle of tracefree endomorphisms of TM which are symmetric with respect to c; thus σ0 satisfies

and hence defines a (weighted) (2,0)-tensor S0 in L4⊗S02⁢T*⁢M, another isomorph of S02⁢T⁢M (in the presence of c).

A conformal Killing (2-)tensor is a section 𝒮0 of S02⁢T⁢M such that the section sym0D⁢𝒮0 of L-2⊗S03⁢T⁢M is identically zero, where D is any Weyl connection (such as the Levi-Civita connection of any compatible metric) and sym0 denotes orthogonal projection onto L-2⊗S03⁢T⁢M inside T*⁢M⊗S2⁢T⁢M≅L-2⊗T⁢M⊗S2⁢T⁢M. Equivalently

for some vector field χ. Taking a trace, we find that

where δD⁢𝒮0 denotes trcD⁢𝒮0, which may be computed, using a conformal frame ei with dual coframe εi, as ∑iDei⁢𝒮0⁢(εi,⋅). Thus 𝒮0 is conformal Killing if and only if

This is independent of the choice of Weyl connection D. On the open set where 𝒮0 is nondegenerate, there is a unique such D with δD⁢𝒮0=0, and hence a nondegenerate 𝒮0 is conformal Killing if and only if there is a Weyl connection D with symD⁢𝒮0=0.

A conformal Killing 2-form is a section ϕ of L3⊗⋀2T*⁢M such that π⁢(D⁢ϕ)=0 (for any Weyl connection D) where π is the projection orthogonal to L3⊗⋀3T*⁢M and L⊗T*⁢M in T*⁢M⊗L3⊗⋀2T*⁢M. It is often more convenient to identify ϕ with a section Φ of L⊗𝔰⁢𝔬⁢(T⁢M) via ϕ⁢(X,Y)=c⁢(Φ⁢(X),Y), where 𝔰⁢𝔬⁢(T⁢M) denotes the bundle of skew-symmetric endomorphisms of TM with respect to c.

B.2 Conformal Killing tensors and complex structures

In four dimensions a conformal Killing 2-form splits into selfdual and antiselfdual parts Φ±, which are sections of L⊗𝔰⁢𝔬±⁢(T⁢M)≅L3⊗⋀±2T*⁢M. Following M. Pontecorvo [43], nonvanishing conformal Killing 2-forms Φ+ and Φ- describe oppositely oriented Kähler metrics in the conformal class, by writing Φ±=ℓ±⁢J±, where ℓ± are sections of L and J± are oppositely oriented complex structures: the Kähler metrics are then g±=ℓ±-2⁢c. Conversely if (g±=ℓ±-2⁢c,J±) are Kähler and D± denote the Levi-Civita connections of g±, then D±⁢(ℓ±⁢J±)=0 so Φ±=ℓ±⁢J± are conformal Killing 2-forms.

The tensor product of sections Φ+ and Φ- of L⊗𝔰⁢𝔬+⁢(T⁢M) and L⊗𝔰⁢𝔬-⁢(T⁢M) defines a tensor Φ+⁢Φ-: as a section of L2⊗Sym0⁢(T⁢M), this is the composite (Φ+∘Φ-=Φ-∘Φ+); as a section of L4⊗S02⁢T*⁢M it satisfies (Φ+⁢Φ-)⁢(X,Y)=c⁢(Φ+⁢(X),Φ-⁢(Y)).

When Φ±=ℓ±⁢J± are nonvanishing, then Φ+⁢Φ-=ℓ+⁢ℓ-⁢J+⁢J- is a symmetric endomorphism with two rank 2 eigenspaces at each point. Conversely if σ0 is such a symmetric endomorphism, we may write σ0=ℓ2⁢J+⁢J- for uniquely determined almost complex structures J± up to overall sign, and a positive section ℓ of L.

Since D is the Levi-Civita connection Dg of the “barycentric” metric g=ℓ-2⁢c, it follows that S0=g(J+J-⋅,⋅) is a Killing tensor with respect to g, i.e., satisfies symDg⁢S0=0 if and only if J+ and J- are integrable and Kähler on average, with barycentric metric g. More generally, we can use this result to characterize, for any metric g in the conformal class and any functions f,h, the case that

is a Killing tensor with respect to g. If θ± are the Lee forms of (g,J±), i.e., D±=Dg±θ±, then we obtain the following more general corollary.

(Obviously when h is identically zero, S is a Killing tensor if and only if f is constant.)

B.3 Conformal Killing tensors and the Ricci tensor

Let

be the tracefree part of the Ricci tensor of a compatible metric g=μg-2⁢c on a conformal n-manifold (M,c). Then, the section 𝒮0g of S02⁢T⁢M, corresponding to the section μg4⁢ric0g of L4⊗S02⁢T*⁢M, is

(where for α∈T*⁢M, g(α♯,⋅)=α)).

The differential Bianchi identity implies that

Hence the following are equivalent:

1. 𝒮0g is a conformal Killing tensor,

2. one has

   symDg⁢𝒮0g=n-2n⁢(n+2)⁢sym(g-1⊗d⁢sg),

3. ricg-2n+2⁢sg⁢g is a Killing tensor with respect to g,

4. for all vector fields X,

   DXg⁢ricg⁡(X,X)=2n+2⁢d⁢sg⁢(X)⁢g⁢(X,X).

Riemannian manifolds (M,g) satisfying these conditions were introduced by A. Gray as 𝒜⁢C⊥-manifolds [24]. Relevant for this paper is the case n=4 and the assumption that ricg has two rank 2 eigendistributions, which has extensively been studied by W. Jelonek [29, 31].

Supposing that g is not Einstein, Corollary 3 implies, as shown by Jelonek, that

is Killing with respect to g if and only if (B.3)–(B.5) are satisfied. Since J± are both integrable, Jelonek refers to such manifolds as bihermitian Gray surfaces. It follows from [7] that both (g,J+) and (g,J-) are conformally Kähler, so that in the context of the present paper, a better terminology would be ambikähler Gray surfaces.

However, the key feature of such metrics is that the Ricci tensor is J±-invariant: as long as J± are conformally Kähler, Proposition 11 applies to show that the manifold is either ambitoric or of Calabi type; it is not necessary that the J±-invariant Killing tensor constructed in the proof is equal to the Ricci tensor ricg.

Jelonek focuses on the case that the ambihermitian structure has Calabi type. This is justified by the global arguments he employs. In the ambitoric case, there are strong constraints, even locally.

B.4 Killing tensors and hamiltonian 2-forms

The notion of hamiltonian 2-forms on a Kähler manifold (M,g,J,ω) has been introduced and extensively studied in [2, 3]. According to [3], a J-invariant 2-form ϕ is hamiltonian if it satisfies

for any vector field X, where X♭=g⁢(X) and σ=trω⁢ϕ=g⁢(ϕ,ω) is the trace of ϕ with respect to ω. An essentially equivalent (but not precisely the same) definition was given in the 4-dimensional case in [2], by requiring that a J-invariant 2-form φ is closed and its primitive part φ0 satisfies

for some smooth function σ. In order to be closed, φ must have the form 32⁢σ⁢ω+φ0.

The relation between the two definitions is straightforward: φ=32⁢σ⁢ω+φ0 is closed and verifies (B.7) if and only if ϕ=φ0+12⁢σ⁢ω satisfies (B.6).

Specializing Corollary 3 to the case when the metric g is Kähler with respect to J=J+ allows us to identify J-invariant symmetric Killing tensors with hamiltonian 2-forms as follows:

B.5 Killing tensors associated to ambitoric structures

We have seen in the previous subsections that there is a link between Killing tensors and ambihermitian structures. We now make this link more explicit in the case of ambitoric metrics.

In the ambitoric situation, the barycentric metric g0 (see Section 4) satisfies

It then follows from Corollary 3 that the (tracefree) symmetric bilinear form g0(I⋅,⋅) (with I=J+∘J-) is Killing with respect to g0. More generally, let g be any (K1,K2)-invariant riemannian metric conformal to g0, so that g can be written as g=h⁢g0 for some positive function h⁢(x,y), where x,y are the coordinates introduced in Section 4. Then

From Corollary 3 again, the symmetric bilinear form S0(⋅,⋅)=hg(I⋅,⋅) is conformal Killing. Moreover, by condition (B.5) in Proposition 3, it can be completed into a Killing symmetric bilinear form S=f⁢g+S0 if and only if the 1-form d⁢h∘I is closed. Since I⁢d⁢x=-d⁢x and I⁢d⁢y=d⁢y, it follows that d⁢h∘I is closed if and only if hx⁢d⁢x-hy⁢d⁢y is closed, if and only if hx⁢y=0; the general solution is h⁢(x,y)=F⁢(x)-G⁢(y), for some functions F,G. Note that the coefficient f⁢(x,y) is determined by d⁢f=-I⁢d⁢h=F′⁢(x)⁢d⁢x+G′⁢(y)⁢d⁢y (see (B.5)), so we can take without loss f⁢(x,y)=F⁢(x)+G⁢(y). Thus, S is Killing, with eigenvalues 2⁢F⁢(x) and 2⁢G⁢(y).

A similar argument shows that any metric of the form g=f⁢(z)⁢g0, where g0 is the barycentric metric of an ambikähler pair of Calabi type and z is the momentum coordinate introduced in Section 3.2, admits a nontrivial symmetric Killing tensor of the form

(and hence with eigenvalues (2⁢f⁢(z),0)).

It follows that there are infinitely many 𝔱-invariant metrics in a given ambitoric conformal class, which admit nontrivial symmetric Killing tensors.

There are considerably fewer such metrics with ambihermitian Ricci tensor. By Proposition 13, these have the form g=h⁢g0, where h⁢(x,y)=(x-y)⁢q⁢(x,y)/p⁢(x,y)2. In order for g to admit a nontrivial symmetric Killing tensor, we must have hx⁢y=0. A calculation shows that this happens if and only if Q⁢(p)=0 (i.e., p⁢(z) has repeated roots). Since p is orthogonal to q, this can only happen if Q⁢(q)≥0 and there are generically (Q⁢(q)>0) just two solutions for p, which coincide if Q⁢(q)=0.